## The Elements of Euclid: With Dissertations Intended to Assist and Encourage a Critical Examination of These Elements as the Most Effectual Means of Establishing a Juster Taste Upon Mathematical Subjects Than that which at Present Prevails |

### From inside the book

Results 1-5 of 97

Page 44

...

...

**joining**BG and FC form two triangles , with all the parts of the fuppofition in the last , accurately represented ; and this not in imagination , but constructed by ourselves ; and the confequences , not made , as was objected to the ... Page 45

... to work until they cease to be the objects of our fenfes . The center of the moon , when in the equinoctial ,

... to work until they cease to be the objects of our fenfes . The center of the moon , when in the equinoctial ,

**joined**by straight lines to the in DISSERTATION 45 II . this pons afinorum, upon a fuppofition that the fourth ... Page 46

...

...

**joined**by straight lines to the two poles of the Earth , would make a very proper isofceles triangle with which the ftudent might finish his fpeculations upon this propofition . СНАР . VII . Concerning indirect demonftrations . INDIRECT ... Page 49

...

...

**joining**CA , CB ; DA , DB and CD ; fo that CD may cut AB or fall beyond the point B. And indeed it is very difficult to under- stand any general propofition , without fome representation of all the different pofitions which the lines ... Page 54

...

...

**joined**already I would defcribc upon it an equilateral triangle producing the fides & c . as directed in the remarks on the third propofition , until the last described circle cut AB in fome fuch point as D ; be- cause the whole ...### Other editions - View all

The Elements of Euclid: With Dissertations Intended to Assist and Encourage ... James Williamson,James Euclid No preview available - 2016 |

### Common terms and phrases

ABCD alfo alſo angle ABC angle BAC angle contained angle equal apply itſelf bafe baſe BC is equal Book certainly circle ABC circumference common notion confequences conft conftruction cut in halves demonftrated deſcribed diſtance drawn equal angles equiangular equilateral equimultiples Euclid exceed faid fame manner fame multiple fame parallels fame ratio fame reaſon fecond fegment fhall fides fimilar fince firſt fome fquare ftraight line BC fuch fuppofe fuppofition given rectilineal given ſtraight line Gnomon greater hath himſelf impoffible infcribed joined lefs leſs let the ftraight magnitudes moſt muſt neceffary parallelogram PROP propofition proportionals purpoſe reader reaſon rectangle contained rectilineal figure remaining angle remaining fides right angles ſame ſay ſhall ſhould ſome ſquare ſtraight line AB ſubject ſuch ſuppoſe taken theſe thoſe tiple triangle ABC underſtand uſe Wherefore becauſe

### Popular passages

Page 3 - Let it be granted that a straight line may be drawn from any one point to any other point.

Page 47 - If there be two straight lines, one of which is divided into any number of parts, the rectangle contained by the two straight lines is equal to the rectangles contained by the undivided line, and the several parts of the divided line. Let...

Page 68 - If a straight line drawn through the centre of a circle bisect a straight line in it which does not pass through the centre, it shall cut it at right angles : and if it cut it at right angles, it shall bisect it.

Page 45 - ABG ; (vi. 1.) therefore the triangle ABC has to the triangle ABG the duplicate ratio of that which BC has to EF: but the triangle ABG is equal to the triangle DEF; therefore also the triangle ABC has to the triangle DEF the duplicate ratio of that which BC has to EF. Therefore similar triangles, &c.

Page 15 - When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands.

Page 86 - When you have proved that the three angles of every triangle are equal to two right angles...

Page 88 - EA : and because AD is equal to DC, and DE common to the triangles ADE, CDE, the two sides AD, DE are equal to the two CD, DE, each to each ; and the angle ADE is equal to the angle CDE, for each of them is a right angle ; therefore the base AE is equal (4.

Page 42 - If four straight lines be proportionals, the rectangle contained by the extremes is equal to the rectangle contained by the means ; And if the rectangle contained by the extremes be equal to the rectangle contained by the means, the four straight lines are proportionals. Let the four straight lines, AB, CD, E, F, be proportionals, viz.

Page 109 - Draw two diameters AC, BD of the circle ABCD, at right angles to one another; and through the points A, B. C, D, draw (17.

Page 8 - GB is equal to E, and CK to F ; therefore AB is the same multiple of E, that KH is of F: But AB...